Converting between Bases

To convert from base $16$ to base $10$, you do the opposite actions, from right to left:

$5\mathrm B92\mathrm C_{16} = 12\cdot16^0 + 2\cdot16^1 + 9\cdot16^2 + 11\cdot16^3 + 5\cdot16^4 = 12 + 32 + 2304 + 45056 + 327680 = 375,084_{16}.$

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The other base conversion method, the division remainder method, is faster and doesn't require you to look up powers of bases. You keep dividing the number by the base (e.g., $2$, $16$, etc.) and take note of the remainders: these will be the digits of the resulting number. This method is best for quickly converting numbers in base $10$ to any base.

Examples: On the following slide, we convert $2619_{10}$ and $1758_{10}$ again to base $2$, but this time using the division remainder method.

In these examples, note how we don't need to memorize any powers of $2$: we just keep dividing by $2$ and notice the remainders we get along the way.